Related work. As mentioned above several aspects of positive coalgebraic logics have been studied in [1]}, [2]}, the concept of posetification and the idea of working in an enriched setting in particular. The key contribution of this work is to dissociate the syntax from the semantics. This reflects the practise of modal logics, where the syntax-building functor \(L\)  is usually not defined directly from the semantics-building functor, but rather from a grammar which is convenient to express certain properties. Graded modal logic for example relies on a syntax which is not obviously related to its semantics. This justifies going beyond the techniques of [2]}. As a consequence, one must also be able to define a semantic natural transformation \(L^{\prime }\mathsf {P}^{\prime }\rightarrow \mathsf {P}^{\prime }T^{\prime }\) , which we do be adapting the boolean semantics. We are also indebted to work on monotone modal logic for the monotone neighbourhood functor, see eg [4]}, [5]}, and on non-monotone modal logic for the (unrestricted) neighbourhood functor as these two cases highlight many of the peculiar features of our approach.
